Peer-to-Peer Networks 15 Self-Organization Christian Schindelhauer - - PowerPoint PPT Presentation

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Oct 24, 2022 127 likes •300 views

Peer-to-Peer Networks 15 Self-Organization Christian Schindelhauer Technical Faculty Computer-Networks and Telematics University of Freiburg Gnutella Connecting Protokoll - Ping Ping participants query for Ping neighbors Ping

SLIDE 1

Peer-to-Peer Networks

15 Self-Organization

Christian Schindelhauer

Technical Faculty Computer-Networks and Telematics University of Freiburg

SLIDE 2

Ping Ping Ping Ping Ping Ping Ping Pong Pong Pong Pong Pong Pong Pong

Gnutella — Connecting

Protokoll
Ping
participants query for

neighbors

are forwarded according

for TTL steps (time to live)

Pong
answers Ping
is forwarded backward on

the query path

reports IP and port adress

(socket pair)

number and size of

available files

SLIDE 3

Degree Distribution in Gnutella

Modeling Large-scale Peer-to-Peer

Networks and a Case Study of Gnutella

Mihajlo A. Jovanovic, Master Thesis,

2001

The number of neighbors is

distributed according a power law (Pareto) distribution

log(#peers with degree d) = c - k log d
#peers with degree d = C/dk

SLIDE 4

Pareto-Distribution Examples

Pareto 1897: Distribution of wealth in the

population

Yule 1944: frequency of words in texts
Zipf 1949: size of towns
length of molecule chains
file length of Unix-system files
….

SLIDE 5

Pareto Verteilung

Discreet Pareto-Distribution for x ∈ {1,2,3,…}
with constant factor
(also known as Riemann´s Zeta-function)
Heavy tail property
not all moments E[Xk] exist
the expectation exists if and only if (iff) α>2
variance and E[X2] exist iff α>3
E[Xk] exists iff α>k+1
Density function of the continuous function for x>x0

SLIDE 6

Indegree and Outdegree of Web-Pages

are described by a power law (Pareto) distribution
Experiments of
Kumar et al 97: 40 millions Webpages
Barabasi et al 99: Domain *.nd.edu + Web-pages in distance 3
Broder et al 00: 204 millions web pages (Scan Mai und Okt. 1999)

SLIDE 7

Connectivity of Pareto Graphs

William Aiello, Fan Chung, Linyuan Lu, A Random Graph

Model for Massive Graphs, STOC 2000

Undirected graph with n nodes where
the probability of k neighbors for a node is pk
where pk = c k-τ for some normalizing factor c
Theorem
For sufficient large n such Pareto-Graphs with exponent τ we observe
for τ < 1 the graph is connected with probability 1-o(1)
for τ > 1 the graph is nont connected with probability 1-o(1)
for 1< τ <2 there is a connected component of size Θ(n)
for 2< τ < 3.4785 there is only one connected component of size Θ(n) and

all others have size O(log n)

for τ >3.4785: there is no large connected component of size Θ(n) with

probability 1-o(1)

For τ >4: no large connected components which size can be described by a

power law (Pareto) distribution

SLIDE 8

Zipf Distribution as a Variant of Power Laws

George Kinsley Zipf claimed
that the frequency of the n most frequent word f(n)
satisfies the equation n f(n) = c.
Zipf probability distribution for x ∈ {1,2,3,…}
with constant factor c only defined for connstant sized sets, since
is unbounded
Zipf distribution relate to the rank
The Zipf exponent α may be larger than 1, i.e. f(n) = c/nα
Pareto distribution realte the absolute size, e.g. the number of

inhabitants

SLIDE 9

Size of towns Scaling Laws and Urban Distributions, Denise Pumain, 2003 Zipf distribution

SLIDE 10

Zipf’s Law and the Internet

Lada A. Adamic, Bernardo A. Huberman, 2002 Pareto Distribution!!

SLIDE 11

Heavy-Tailed Probability Distributions in the World Wide Web Mark Crovella, Murad, Taqqu, Azer Bestavros, 1996

SLIDE 12

Small World Phenomenon

Milgram’s experiment 1967
60 random chosen participants in Wichita, Kansas had to

send a packet to an unknown address

They were only allowed to send the packet to friends
likewise the friends
The majority of packtes arrived within six hops
Small-World-Networks
are networks with Pareto distributed node degree
with small diameter (i.e. O(logc n))
and relatively many cliques
Small-World-Networks
Internet, World-Wide-Web, nervous systems, social networks

SLIDE 13

How do Small World Networks Come into Existence?

Emergence of scaling in random networks, Albert-Laszlo

Barabasi, Reka Albert, 1999

Preferential Attachment-Modell (Barabasi-Albert):
Starting from a small starting graph successively nodes are inserte with

m edges each (m is a parameter)

The probability to choose an existing node as a neighbor is proportional

to the current degree of a node

This leads to a Pareto network with exponent 2,9 ± 0,1
however cliques are very seldom
Watts-Strogatz (1998)
Start with a ring and connections to the m nearest neighbors
With probability p every edge is replaced with a random edge
Allows continuous transition from an ordered graph to chaos
Extended by Kleinberg (1999) for the theoretical verification of

Milgram‘s experiment

SLIDE 14

Analyzing Gnutella

Modeling Large-scale Peer-to-Peer Networks and

a Case Study of Gnutella

Mihajlo A. Jovanovic, 2001

SLIDE 15

Analyzing Gnutella

Modeling Large-scale Peer-to-Peer Networks and

a Case Study of Gnutella

Mihajlo A. Jovanovic, 2001
Comparison of the characteristic path length
mean distance between two nodes